Factorization of entire symmetrical functions of exponential type

A. B. Shishkin

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 16 (2016) no. 1, pp. 42-68

Résumé

Let $\pi$ be an entire function of minimal type of order $1$. The entire function $F$ is called $\pi$-symmetric if it is represented in the form of a composition $f\circ\pi$, where the $f$ is an entire function. The article deals with the following question. Can we present every $\pi$-symmetric function of exponential type as a product of two functions with a close growth, each of which is itself an entire $\pi$-symmetric function? This question is answered in the affirmative, but under certain restrictions on for the subordinate function $\pi$. For example, an entire function of completely regular growth at proximate order $\rho(r) \approx\rho\in(0;1)$ with constant positive indicator is subject to these restrictions. Other examples relate to the reversibility of the entire function in the circles of constant radius whose centers lie outside some exceptional set.